Relative energy (or entropy) inequality in the context of compressible Navier-Stokes system was first introduced by Germain in [1]}, and it was used as a tool in proving the weak-strong uniqueness property for a sub-class of finite energy weak solutions which have have a \(L_t^p L_x^q\) -integrable gradient of density (for certain \(p,q>1\) ). Note that the latter condition is quite unnatural and existence of weak solutions within this class is non-existant in the literature. Later, Feireisl, Novotný and Sun in [2]} introduced suitable weak solutions - a sub-class of finite energy weak solutions that satisfy the relative energy inequality, then proved their existence and showed the weak-strong uniqueness within this class, thus removing the additional regularity on density imposed in [1]}. Finally, Feireisl, Novotny and Jin in [4]} introduced the relative entropy in the class of finite energy weak solutions and gave a definitive answer to the weak-strong uniqueness question, proving that all finite energy weak solutions are in fact suitable weak solutions and thus have the weak-strong uniqueness property. Here, we extend the notions of relative entropy and relative energy inequality to our fluid-structure interaction model.
